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LO(\)-l)(t) \)= 1 for some unique hyperfunctions tpl, ••• ,tpr are called the boundary values of u. on {t = O}. tpl, ••• ,tpr The generalization to the equation with regular singularities by Kashiwara and Oshima takes a similar course, reflected in the two theorems below. 7) s \I(x) !! ) ';:i'I (H) supp u (Hi) Pu M + = \I = 1, ••• , r . u e ~(M) unique hyperfunction ° =° t > be given by sl,···,sr Suppose in the local coordinates. 2 b s\l is important. 1. 4 and p. 175) Assume that s\I(x) - a neighborhood of A\I(x,Dx,D t ) - to 1 on - 1\ (\1= 8\1' 1\, (x) ~ Z for all \) l' \1', X eN.

1, the material of which is classical (see for instance Coddington and Levinson raj), the theory of this chapter is mainly due to M. Kashiwara and T. Oshima raj. 6 (i) are all taken from there. 6 (ii) is from Oshima and Sekiguchi raj. For different approaches to partial differential equations with regular singularities we refer to Harish-Chandra [g J,. Cas selman and Mili~i~ raJ, Wallach [cJ and the appendix section of Knapp [bJ. In Oshima [eJ a simpler but not yet as powerful theory is presented.

A] p. 276, Definition 1. 3. 3 (cf. also Sato[d]) a more intrinsic definition of a sheaf t; on R S*M is given: 4 =UnFI S*M(TT-1 (9) a , where a is the antipodal map Ff s*M -> Pi s*M • In loco cit. p. 6), that this sheaf is flabby. p. 8). For each open set From loco cit. ) i 1. ~ 1. 1. 1 fi E A-(V). I f. e~(V) Let 1. 1. 3(ii) (cf. Morimoto [a], Theorem 16). Let N be a real analytic submanifold of to define the restriction to M. Then it is possible N of a hyperfunction on M provided its singular spectrum satisfies a certain condition.

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